Student standardized-test scores at an award-winning D.C. school dropped dramatically in 2011 after the principal tightened security out of concern about possible cheating, according to a new “Frontline” television documentary to be broadcast Tuesday.

The hour-long program raises questions about whether District officials have adequately investigated persistent suspicions that public school employees may have tampered with tests during the tenure of former schools chancellor Michelle A. Rhee.

Adell Cothorne was principal of the District’s Noyes Education Campus for one year, in 2010-11. She told “Frontline” that just after students took a midyear practice version of the city’s annual standardized test, she stumbled upon three staff members sitting late at night in a room strewn with more than 200 test booklets.

One of the adults was at a desk, holding an eraser. The other two sat at a table, booklets open before them.

“One staff member said to me, in a lighthearted sort of way, ‘Oh, principal, I can’t believe this kid drew a spider on the test and I have to erase it,’ ” Cothorne told filmmakers, offering the first such direct testimony about potential tampering with answer sheets in D.C. schools.

Cothorne told “Frontline” that she reported the incident to the central office, but to her knowledge nothing was done. School system officials said Friday that without having seen the documentary, they could not comment on Cothorne’s allegations.

“Broadly speaking, reports about testing impropriety are taken very seriously,” D.C. Public Schools spokeswoman Melissa Salmanowitz wrote in an e-mail. “We have investigated and taken appropriate action for every instance reported to us.”

The “Frontline” film, by education reporter John Merrow, examines Rhee’s record in the District, where her aggressive reforms between 2007 and 2010 turned her from a relative unknown into a polarizing edu-celebrity, both applauded and criticized for her unapologetic approach to fixing the District’s long-troubled schools.

Much of the film, “The Education of Michelle Rhee,” draws on footage previously broadcast in a dozen of Merrow’s “PBS NewsHour” reports. It chronicles the failings of the school system and the chancellor’s efforts to turn it around by closing schools with low enrollment and firing ineffective teachers, principals and central office workers.

In one well-known scene, Rhee fires a principal on tape, telling him in front of the film crew that his leadership had been “completely unimpressive.”

“This is not about giving people jobs, or ensuring that people can maintain their jobs. This is about educating children,” Rhee told Merrow, voicing the no-excuses approach to school reform — and the impatience with the city’s dismal academic results — that won her admiration around the country.

One of Rhee’s signature moves was turning the D.C. Comprehensive Assessment System, the city’s annual standardized test, into a high-stakes event for teachers and principals. For the first time, their jobs and their pay depended upon raising student scores.

The District’s test scores did rise during Rhee’s tenure, including a particularly dramatic jump after her first year in office.

Rhee left office in 2010 after then-Mayor Adrian Fenty, who had hired her, lost his bid for reelection. She then launched Students First, a national lobbying organization to spread the reforms she championed in Washington.

But in 2011, USA Today published an investigation that raised questions about the validity of the District’s test scores — and, by proxy, about the effectiveness of Rhee’s reforms.

The newspaper’s report revealed an unusually high number of wrong-to-right erasures on students’ answer sheets at more than 100 D.C. schools between 2008 and 2010. Such erasure rates aren’t proof of cheating, but they are signals of potential tampering.

Current Schools Chancellor Kaya Henderson asked D.C. Inspector General Charles J. Willoughby to investigate further. Willoughby reported in August that he’d found no evidence of answer-sheet tampering, a conclusion that Henderson said should finally put cheating allegations to rest.

The “Frontline” documentary, however, suggests the inspector general’s investigation may have been incomplete.

The 17-month probe focused on just one school: Noyes, which was named a National Blue Ribbon School in 2009 after students made impressive gains on reading and math tests. It also twice won an award from Rhee that brought cash bonuses for staff, and it had some of the highest erasure rates in the city.

Investigators found some test-security problems at Noyes but no evidence of answer-sheeting tampering. Based on those findings, they decided not to examine other schools.

But Cothorne, the former principal who alleges that she saw staff members after hours with erasers and test booklets, said investigators never interviewed her.

“My speculation: They didn’t want to hear what I had to say,” she told “Frontline.”

The inspector general’s office declined “Frontline’s” request for an interview about its investigation, saying the report speaks for itself. The office also declined The Post’s request for comment Friday.

DCPS spokeswoman Salmanowitz said school officials remain confident in the inspector general’s work.

Salmanowitz also said records show that Cothorne did not mention the incident when she was interviewed in March 2011 by Caveon, a company retained by the school system to examine potential cheating over several years.

Cothorne denied that she was interviewed by Caveon, but said as a new principal in D.C. schools, she was scared about speaking up.

Cothorne told the “Frontline” filmmakers that when she arrived at Noyes in 2010, she noticed that students’ academic abilities didn’t match their test scores.

Then she said she found the three staff members with test booklets from the midyear D.C. Benchmark Assessment System, a standardized exam that tracks students’ progress toward the end-of-year test — the one that matters for teachers’ and principals’ jobs.

Why would a teacher be motivated to inflate practice-test scores? Perhaps to make high end-of-year scores more believable, Cothorne said in an interview Friday.

Cothorne told “Frontline” that she tightened test security. On the end-of-the-year test, math and reading scores dropped more than 25 percentage points from the year before. The principal left Noyes at the end of that school year and opened a cupcake shop in Ellicott City.

Several other D.C. schools that made impressive test-score gains under Rhee saw double-digit declines after test security was tightened.

Asked for her reaction, Rhee told Merrow that such large swings should be investigated. There may have been some problems, she allowed — but they were isolated.

“I can point to . . . dozens and dozens of schools where, you know, they saw very steady gains,” Rhee said, “or even saw some dramatic gains that were maintained.”

She told Merrow that she was sorry to leave the District.

“I lost the job that I loved,” she said. “The work that we’re doing right now with Students First is important. Would I rather be in D.C. as the chancellor? Absolutely.”

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